Properties

Label 2-17e2-1.1-c9-0-29
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 40.6·2-s − 102.·3-s + 1.13e3·4-s − 769.·5-s + 4.18e3·6-s + 1.56e3·7-s − 2.53e4·8-s − 9.08e3·9-s + 3.12e4·10-s − 1.70e4·11-s − 1.17e5·12-s + 1.41e5·13-s − 6.36e4·14-s + 7.92e4·15-s + 4.48e5·16-s + 3.68e5·18-s − 7.43e5·19-s − 8.75e5·20-s − 1.61e5·21-s + 6.91e5·22-s − 1.55e6·23-s + 2.61e6·24-s − 1.36e6·25-s − 5.76e6·26-s + 2.96e6·27-s + 1.78e6·28-s + 5.73e6·29-s + ⋯
L(s)  = 1  − 1.79·2-s − 0.733·3-s + 2.22·4-s − 0.550·5-s + 1.31·6-s + 0.246·7-s − 2.19·8-s − 0.461·9-s + 0.988·10-s − 0.350·11-s − 1.63·12-s + 1.37·13-s − 0.442·14-s + 0.404·15-s + 1.71·16-s + 0.828·18-s − 1.30·19-s − 1.22·20-s − 0.181·21-s + 0.629·22-s − 1.15·23-s + 1.60·24-s − 0.696·25-s − 2.47·26-s + 1.07·27-s + 0.547·28-s + 1.50·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.2525577056\)
\(L(\frac12)\) \(\approx\) \(0.2525577056\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 40.6T + 512T^{2} \)
3 \( 1 + 102.T + 1.96e4T^{2} \)
5 \( 1 + 769.T + 1.95e6T^{2} \)
7 \( 1 - 1.56e3T + 4.03e7T^{2} \)
11 \( 1 + 1.70e4T + 2.35e9T^{2} \)
13 \( 1 - 1.41e5T + 1.06e10T^{2} \)
19 \( 1 + 7.43e5T + 3.22e11T^{2} \)
23 \( 1 + 1.55e6T + 1.80e12T^{2} \)
29 \( 1 - 5.73e6T + 1.45e13T^{2} \)
31 \( 1 + 4.58e6T + 2.64e13T^{2} \)
37 \( 1 + 5.48e6T + 1.29e14T^{2} \)
41 \( 1 - 1.81e7T + 3.27e14T^{2} \)
43 \( 1 - 2.79e7T + 5.02e14T^{2} \)
47 \( 1 - 2.65e7T + 1.11e15T^{2} \)
53 \( 1 - 7.58e7T + 3.29e15T^{2} \)
59 \( 1 + 4.75e7T + 8.66e15T^{2} \)
61 \( 1 + 1.40e8T + 1.16e16T^{2} \)
67 \( 1 + 7.10e7T + 2.72e16T^{2} \)
71 \( 1 + 7.14e7T + 4.58e16T^{2} \)
73 \( 1 + 3.40e8T + 5.88e16T^{2} \)
79 \( 1 - 3.20e8T + 1.19e17T^{2} \)
83 \( 1 - 3.26e8T + 1.86e17T^{2} \)
89 \( 1 + 2.75e8T + 3.50e17T^{2} \)
97 \( 1 + 6.65e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51313149184397818935291610887, −9.093492250524772590876587495145, −8.379309535211148962382480222045, −7.72860769235130687373566196311, −6.48489499017694220871561330046, −5.82254246134836961016913087185, −4.11221775810939446637639854847, −2.57149471630148248317117307014, −1.37457233688764081819075192066, −0.32521229414962724500738451347, 0.32521229414962724500738451347, 1.37457233688764081819075192066, 2.57149471630148248317117307014, 4.11221775810939446637639854847, 5.82254246134836961016913087185, 6.48489499017694220871561330046, 7.72860769235130687373566196311, 8.379309535211148962382480222045, 9.093492250524772590876587495145, 10.51313149184397818935291610887

Graph of the $Z$-function along the critical line